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Adding and Subtracting Negative Integers as Temperatures

Date: 01/08/2004 at 00:52:50
From: Jonah
Subject: word problems with negative integers

If it's -20 degrees outside and it decreases 3 degrees what
temperature is it?  I know the answer is -23 degrees but it seems like
it should be -17 degrees.  When do you have to change the sign into
addition?



Date: 01/08/2004 at 12:13:38
From: Doctor Peterson
Subject: Re: word problems with negative integers

Hi, Jonah.

It may help to look at an actual thermometer. It will look something 
like this:

    |
    + 40
    |
    + 30
    |
    + 20
    |
    + 10
    |         ^
    + 0      / \
    |         |
    + -10     | warmer (increasing temperature)
    |         |
    + -20 ----+
    |         |
    + -30     | colder (decreasing temperature)
    |         |
    + -40    \ /
    |         V

Suppose it was -20 degrees an hour ago, and now it has gone down 10 
degrees.  As it goes down, it will get farther away from 0 (since it 
is below zero to start with), so it will decrease from -20 degrees 
to -30 degrees!  The number part (the 30, which is the "absolute 
value" of the temperature) increases because we are going away from 
zero while the temperature itself decreases (getting colder).  If the 
temperature changed to -10, it would be warmer, not colder; that would
be an increase in temperature.

Does that make sense?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 01/08/2004 at 12:09:15
From: Doctor Rick
Subject: Re: word problems with negative integers

Hi, Jonah.

It's easier to understand the correct answer if you look at a
thermometer.  If it had every degree labeled, portions of it would 
look like this:

   5 | |
   4 | |
   3 | |
   2 | |
   1 | |
   0 | |
  -1 | |
  -2 | |
  -3 | |
  -4 | |
  -5 | |
  -6 | |
.........
 -18 | |
 -19 | |
 -20 |-|  *
 -21 | |  | down 3
 -22 | |  V
 -23 | |  *
 -24 | |
 -25 | |

If you're at -20 and you go DOWN (decrease) 3 degrees, you get to -23.

Now, to do it mathematically instead of looking at the thermometer, 
we do this:

  -20 - 3

which is the same as

  -20 + (-3)

Subtracting is always the same as adding the opposite.  Now, if you're
familiar with the distributive property, we can remember that the
negative sign means to multiply by -1, and we can do that
multiplication just once:

  (-1*20) + (-1*3) = -1*(20 + 3)
                   = -1*23
                   = -23

In other words, the rule for adding two negative numbers is to add 
the numbers without the negative signs, then make the result negative.

If we wanted to INCREASE the temperature 3 degrees, we would do this:

  -20 + 3

I can again factor out a -1 like this:

  (-1*20) + (-1*-3)

because 3 divided by -1 is -3.  Then combine the -1's with the 
distributive property:

  -1(20 + -3) = -1(20 - 3)
              = -1*17
              = -17

A rule to remember here is that if you are adding two numbers with 
DIFFERENT signs, you subtract the smaller from the bigger (ignoring 
the negative sign), and give the answer the sign of the bigger 
number.  In this problem, -20 + 3, we look at 20 and 3, subtract 3 
from 20 to get 17, then since the bigger number (20) has the negative 
sign, we make the answer negative: -17.

I hope this helps you!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Temperature
Middle School Negative Numbers
Middle School Temperature

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